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A069904 Number of prime factors of n-th triangular number (with multiplicity).

%I A069904 #32 Aug 05 2019 18:23:16
%S A069904 0,1,2,2,2,2,3,4,3,2,3,3,2,3,5,4,3,3,3,4,3,2,4,5,3,4,5,3,3,3,5,6,3,3,
%T A069904 5,4,2,3,5,4,3,3,3,5,4,2,5,6,4,4,4,3,4,5,5,5,3,2,4,4,2,4,8,7,4,3,3,4,
%U A069904 4,3,5,5,2,4,5,4,4,3,5,8,5,2,4,5,3,3,5,4,4,5
%N A069904 Number of prime factors of n-th triangular number (with multiplicity).
%H A069904 Antti Karttunen, <a href="/A069904/b069904.txt">Table of n, a(n) for n = 1..20000</a>
%F A069904 a(n) = A001222(A000217(n)).
%F A069904 From _Antti Karttunen_, Oct 07 2017: (Start)
%F A069904 a(n) = (A001222(n)+A001222(n+1))-1.
%F A069904 a(n) = A001222(A278253(n)). (End)
%F A069904 From _Alois P. Heinz_, Aug 05 2019: (Start)
%F A069904 a(n) = 2 <=> n in { A164977 }.
%F A069904 a(n) = 3 <=> n in { A108815 }.
%F A069904 a(n) = 4 <=> n in { A114435 }.
%F A069904 a(n) = 5 <=> n in { A114436 }.
%F A069904 a(n) = 6 <=> n in { A114437 }.
%F A069904 a(n) = 7 <=> n in { A240527 }.
%F A069904 a(n) = 8 <=> n in { A240528 }.
%F A069904 a(n) = 9 <=> n in { A240529 }.
%F A069904 a(n) = 10 <=> n im { A101745 }. (End)
%e A069904 A000217(8) = 8*(8+1)/2 = 36 = 2*2*3*3, therefore a(8) = 4.
%t A069904 Array[Plus@@Last/@FactorInteger[ #*(#+1)/2]&,33] (* _Vladimir Joseph Stephan Orlovsky_, Feb 28 2010 *)
%o A069904 (PARI) A069904(n) = bigomega((n*(n+1))/2); \\ _Antti Karttunen_, Oct 07 2017
%Y A069904 Cf. A069901, A069902, A069903, A092405, A101745, A108815, A114435, A114436, A114437, A164977, A240527, A240528, A240529, A278253.
%K A069904 nonn
%O A069904 1,3
%A A069904 _Reinhard Zumkeller_, Apr 10 2002